A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games

By Ian M. Mitchell, Alexandre M. Bayen, Claire Jennifer Tomlin. 2005

1,261 citations recorded by the source corpus.

Subfields: Aerospace Engineering, Computational Theory and Mathematics, Statistics and Probability, Game & Decision Theory

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Citations in the corpus

Concrete Problems in AI Safety (2016, background): “For the latter challenge, there is also relevant work based on reachability analysis [91, 98] and robust policy improvement [160], which provide potential methods for deploying conservative policies in situations of uncertainty; to our knowledge, this work has not yet been combined with methods for ”
Certified Defenses against Adversarial Examples (2018, methodology): “In robotics, certiﬁcates of stability are routinely used both for safety veriﬁcation (Lygeros et al., 1999; Mitchell et al., 2005) and controller synthesis (Ba¸sar & Bernhard, 2008; Tedrake et al., 2010).”
Multiplayer reach-avoid games via low dimensional solutions and maximum matching (2014, methodology): “We first describe the HJ reachability framework for solving differential games with arbitrary terrain, domain, obstacles, target set, and player velocities based on [15], [22], [5].”
Reducing Conservativeness in Safety Guarantees by Learning Disturbances Online: Iterated Guaranteed Safe Online Learning (2012, methodology): “1) Safety via HJI Reachability Analysis: HJI reachability analysis [10] assumes a system model of the form ẋ = f(x, u, d), where x ∈ Rn is the system state, u is the control input to the system, and d is the disturbance to the system, with u and d bounded to be in some sets U and D, respectively.…”
Safe Control under Uncertainty (2015, methodology): “For instance, designing controllers under reachability analysis is a well-studied method that allows specifying safety and reachability properties [29, 30].”
Reach-avoid problems with time-varying dynamics, targets and constraints (2014, background): “While computationally intensive, HamiltonJacobi approaches are practically appealing nowadays due to the availability of modern numerical tools such as [14, 17, 19, 23], which are able to solve the associated equations for these problems when the dimensionality is low.”
A General Safety Framework for Learning-Based Control in Uncertain Robotic Systems (2017, methodology): “dpxq P D̂pxq @x P R), or at least everywhere in the constraint set K [25, 39].”
Cooperative pursuit with Voronoi partitions (2016, neutral): “ple, be obtained by solving the corresponding Hamilton-JacobiIsaacs (HJI) partial differential equation (PDE) [4, 11, 18, 21].”
Reachability-based safe learning with Gaussian processes (2014, background): “modified terminal value HJI partial differential equation [15].”
A path defense approach to the multiplayer reach-avoid game (2014, methodology): “Modern numerical tools such as [15], [16], [17] are able to solve the HJI PDE when the dimensionality of the problem is low.”
Provably safe and robust learning-based model predictive control (2011, positive): “The main insight of LBMPC is that performance and safety can be de-coupled in an MPC framework by using reachability tools [4,14,56,23,5,69,52].”
LQR-trees: Feedback Motion Planning via Sums-of-Squares Verification (2010, methodology): “Computing reachable sets of uncertain nonlinear system has been explored using a dynamic game framework [19].”
Measuring and avoiding side effects using relative reachability (2018, background): “A similar argument applies to reachability analysis methods (Mitchell et al., 2005; Gillula and Tomlin, 2012; Fisac et al., 2017), which require that a safe region must be reachable by a known conservative policy: the agent would be penalized if another agent or an environment event make the safe…”
Set Propagation Techniques for Reachability Analysis (2021, neutral): “Some approaches rely on optimal control theory to show that the reachable set of a system corresponds to the zero sublevel set of the solution of a Hamilton-Jacobi partial differential equation [26, 27].”
Hamilton–Jacobi Formulation for Reach–Avoid Differential Games (2009, methodology): “Based on the analysis of [33], (8) implies that the trajectory of the augmented system visits only the subset of the states visited by the trajectory of the original system in the time interval .”
Time-optimal path planning in dynamic flows using level set equations: theory and schemes (2014, methodology): “HJB equations also form the basis of several approaches to compute the reachability fronts in areas of game theory and differential games (Mitchell et al. 2005; Bokanowski et al. 2010).”